I'm trying to get a grasp on infinitives and have the following question:

Suppose you have the following english sentence in Greek:

I need to appear in court.

With "to appear" being the greek infinitive.

1)
What would the differences in meaning be in this sentence if it was a present infinitive compared to a future infinitive?

2)
What would the differences in meaning be in this sentence if it was a aorist infinitive compared to a future infinitive?

I can make a guess at number 1. Present would show an ongoing need to appear in court (perhaps they are a lawyer) vs simply the need to appear in court at some future date.

I can't even make a guess at number two. Does the idea of a future infinitive even make sense? Obviously if one needs to do something, then they are not doing it now and it has to be future regardless of tense?

The future can also have the (now I know why some English grammar books might be a good idea) 'final' aspect. That's what the Germans call it anyway .
Like:
I go to court so that I can appear.
But in this sentance, well uh... I can't think of a Greek verb need with infinitive only with gen.
I know nothing about English- Greek translation at all, though.
I need to appear in court in future. Lol, think that's what the difference is if you have a future infinitive . Let's just wait for William or Skylax or anyone else good at Greek to reply...

Aorist is one point in time, so depending on wether you need to be in court at a certain point in time or over a period of say months you use either aorist (and because it's infinitive it's not past, it's only really past if it's in the indicative) or present.

1)What would the differences in meaning be in this sentence if it was a present infinitive compared to a future infinitive?

In this sort of construction the future infinitive would not be used at all; it's mostly used in indirect statements - "he said that the general will go." (Normally better English is "the general would go.")

2)

What would the differences in meaning be in this sentence if it was a aorist infinitive compared to a future infinitive?

I can make a guess at number 1. Present would show an ongoing need to appear in court (perhaps they are a lawyer)

That is correct. The aorist indicative means past time, but the infinitive, the imperative, the participle and the subjunctive and optative mean single action (or thought of as single event: for example, in the middle of a war, a battle is ongoing; 30 years later, the battle appears as one event).

So, present: I need to appear in court (duration implied);aorist: I need to appear in court (single event).

I can't even make a guess at number two. Does the idea of a future infinitive even make sense?

Yes, but Greek doesn't do it that way.

Obviously if one needs to do something, then they are not doing it now and it has to be future regardless of tense?

That also makes sense, and that's what Greek does. So in a sentence like "I want to go to Greece" the future is very strongly implied, but the grammar is going to use a present or aorist infinitive.